Question

The \(\Delta H\) and \(\Delta S\) values of a reaction are \(400\,\mathrm{kJ\,mol^{-1}}\) and \(200\,\mathrm{J\,K^{-1}\,mol^{-1}}\) respectively which are constant over a wide range of temperature. The temperature above which the reaction will be spontaneous is

• A. \(2\,K\)
• B. \(400\,K\)
• C. \(2000\,K\)
• D. \(800\,K\)

Hint:

For reactions with \[ \Delta H\gt 0,\qquad \Delta S\gt 0 \] spontaneity is achieved at high temperatures. Use \[ T=\frac{\Delta H}{\Delta S} \] to find the minimum temperature for spontaneity.

Answer 1

The Correct Option is C

Step 1: Recall the spontaneity condition.
A reaction is spontaneous when the Gibbs free energy change is negative, $\Delta G \lt 0$.

Step 2: Write the Gibbs equation.
\[ \Delta G = \Delta H - T\Delta S \]

Step 3: Find the borderline temperature.
The reaction just turns spontaneous when $\Delta G = 0$, so
\[ T = \frac{\Delta H}{\Delta S} \]

Step 4: Match the units.
Convert $\Delta H = 400\,kJ = 400000\,J$ per mol, and $\Delta S = 200\,J\,K^{-1}\,mol^{-1}$.

Step 5: Put in the numbers.
\[ T = \frac{400000}{200} = 2000\ K \]

Step 6: State the answer.
Above $2000\,K$ the term $T\Delta S$ beats $\Delta H$, so $\Delta G$ becomes negative and the reaction is spontaneous.
\[ \boxed{2000\ K} \]